My hope of a new version each month has clearly not worked out! But here is one for April, just before the month ends. As usual, a random selection of corrections and minor improvements have been dealt with, and many more remain to do.

The April version is posted at the usual place (the April 29, 2015 version).


I am going to attempt to put up a version every month, in order to ensure that I continue to spend a little time tidying things up every so often.  So the January version is now  posted at the usual place (the January 29, 2015 version, also in e-reader format).

The list of things to do has shrunk, but there is a good deal still to do.  I have had many useful detailed comments to later digest and then include from Gurbir Dhillon, Tony Feng, Lisa Sauermann, and Jesse Silliman; and Benjamin Ljundberg, Chandrasekhar Raju, and Scott Zhang.


The last version posted was in 2013. I wanted to get the next version out in 2014, and time is running out!  So here it is, posted at the usual place (the Dec. 30, 2014 version).  (There is also a version suited for an e-reader — thanks to Jack Sherk asking for it, and explaining to me how to make it.)

In this 2014 post, I can’t help mentioning the passing of Grothendieck.   I was more moved than I expected to hear the news and feel the ripples in the mathematical community.  It feels strange that he is now a historical figure, even though he had walked away from his unfinished cathedral long before I was even aware of its existence.

The notes have continued to evolve around the edges, although the material is stable. Please continue to give me corrections and suggestions!  There are few sections that need tender loving care; I mention them on the front.  There are many other things on my to-do list as well, including many comments you’ve made.

The notes now have a tentative title (The Rising Sea:  Foundations of Algebraic Geometry).  The phrase is due to Grothendieck, translated by Colin McLarty; it is the title of Daniel Murfet’s wonderful blog.

The index is in the process of being made.  (No need to give me specific comments until it has converged!)

Finally, if you would like notes on commutative algebra that are very much from the same point of view as these notes, you may enjoy Andy McLennan’s notes, available here.  It also contains an (explicated) English translation of Serre’s epochal FAC, and all the algebra needed as background.

The course webpage is here. The course notes are here.

A new version is now posted at the usual place (the June 11, 2013 version).   The main reason for this post is to have a reasonably current version on the web.  Thanks to many people for helpful comments — most recently, a number of comments from János Kollár, Jeremy Booher, Shotaro (Macky) Makisumi, Zeyu Guo, Shishir Agrawal, Bjorn Poonen, and Brian Lawrence (as well as many people posting here, whose names I thus needn’t list).

There are a large number of very small improvements, and I’ll list only a few in detail.  I’ve replaced the proof of the Fundamental Theorem of Elimination Theory (Theorem 7.4.7).  (This new proof is much more memorable for me.  It’s also shorter and faster, and generally better.  It was the proof I was trying to remember, which I heard in graduate school.  I couldn’t find it in any of the standard sources, and reproduced it from memory.  But it must be in a standard source, because it is certainly not original with me!)   Jeremy Booher’s comments have led to the completion discussion being improved a lot.

I had earlier called \rm{Spec} A, where A is the subring k[x^3, x^2, xy, y] of k[x,y], the “knotted plane”, to suggest the picture that it was a plane, with the origin somehow “knotted” or “pinched”.  János Kollár pointed out that “knotted” is misleading, because it is not in any obvious way knotted.  I’ve changed this to “crumpled plane”, but this isn’t great either.  I’m now thinking about “pinched plane”.  Does anyone have a good suggestion for a name for this important example?

Still to do:  To repeat my comments from the previous post, as usual, the figures, index, and formatting have not yet been thought about.  My to-do list is quite short, so please complain about anything and everything (except figures, index, and formatting).  (Perhaps in the next posted version, I may have “TODO” appearing in the text indicating where there is still work in progress.)

Finally, just for fun, here is a picture of the two rulings on the quadric surface, from the Kobe skyline.  (It is clear why algebraic geometry is so strong in Japan!)

Kobe port

Kobe port (click to enlarge)

A new version is now posted at the usual place (the Mar. 23, 2013 version).   There’s not much to report.  I’ve responded to the advice you’ve given in the previous post, done the bibliography (so in particular, you are free to criticize it), given a little more geometric motivation for completions following the advice of Andrei, and responded to more suggestions sent by email, and by in this year’s Stanford reading group.

Still to do:  As usual, the figures, index, and formatting have not yet been thought about.  (Again:  the bibliography is off this list!  I’ve gotten advice from a number of people on the index, notably Rob Lazarsfeld, and I have at least some idea of how I want to proceed.)  I have a to-do list of precisely 50 items.  (Perhaps in the next posted version, I may have “TODO” appearing in the text indicating where there is still work in progress.)

Questions for you:  it may soon come time to make figures.  Do you have recommendations on how to make pictures?  You might guess my criteria:  I would like to make them look reasonably nice, but they needn’t be super-fancy; and the program should be easy for me to use (I am a moron about these things), and ideally cheap or free.  I’ve used xfig in the past in articles (and the figures currently made), and like it a lot, but I’ve found it imperfect for more complicated figures (with curvy things), and a little primitive for somewhat complicated figures.  I’m remotely considering finding someone who is good at this, and seeing how much they might cost.

Added April 26, 2013:  Charles Staats sent me this beautiful picture of a blow-up.  (I currently haven’t imported it into the file, because for compiling reasons the file goes through dvi, not pdflatex; but don’t bother telling me how to fix this, as I can always ask later if it becomes urgent.)  Caution:  it didn’t view well on my browser; you may want to download it to view it properly.

(Update May 15, 2013:  at this point, to my amazement, all of my questions have been answered, although more good answers to the rest would be most welcome.  I have learned a great deal of neat stuff I should have known long before by asking these questions!)

A new version is now posted at the usual place (the Feb. 25, 2013 version).  There are many small improvements and patches, but no important changes.  Because I want to give you something new to look at, here is a newly added diagram of chapter dependencies.  I was a little surprised by what it showed.

Diagram of Chapter Dependencies

Diagram of Chapter Dependencies

The “to-do” list of things to be worked on is now at about 100 items.   With the exception of formatting, figures, the bibliography, and the index, I continue to be very interested in hearing of any suggestions or corrections you might have, no matter how small.  We are clearly nearing the endgame.

I now want to ask advice on a number of issues all at once.  These are mostly small things, and are along the lines of “what is the best reference to point learners to on this topic”.  Some of the questions are “unimportant”, in the sense that I doubt they will affect my exposition (although they may be important in some larger sense).  For those questions relating to some particular part of the notes, I will give the section number.  Please feel free to respond by email or in the comments.  Here we go!

Peter Johnson strongly preferred using “fibre” instead of “fiber”.  Does anyone else feel strongly?

1.4.1    Latex question:  how do you get the \varprojlim subscript in the right place? (answer here)

1.6.12   (unimportant) Do right-exact functors always commute with colimits?  (For example, M \otimes  \cdot commutes with direct sums, which is what we use, but that is easy to check directly.) (answer here)

5.4.M   Can anyone get this exercise (that, basically, says that normality descends under finite field extensions)?  I think it should be gettable, but not easy, but I’ve had clues that this is harder than I thought.  I want to be sure I have the level correctly gauged.  (Feel free to respond by email if you get stuck.)  (two positive response received so far, including this one)

5.4.N   {\mathbb{Z}}[\sqrt{-5}] is not a unique factorization domain, but its Spec can be covered with 2 (distinguished) affine subsets, each of which are Specs of UFD’s.  Is there some good reference for this?  (Presumably it becomes a UFD upon inverting either 2 or 3, but I can’t see why this is the case.   And of course I don’t just want to know what is true; I’d like a reference for why it is true.)  Added later:  I should also have added, is there a well-loved reference that shows that the class group of {\mathbb{Z}}[\sqrt{-5}] is \mathbb{Z}/2(answer here

6.3.K A compact complex variety can have only one algebraic structure.   What is a reference?  (A number of sources mention this fact, but I want an actual proof.)  On a related point, in 10.3:  A variety over \mathbb{C} is proper if and only if it is compact in the “usual” topology.  What is a reference?  (answer here and here)

6.7  In this section, I mention the Schubert cell decomposition of the Grassmannian.  The key idea is that any k-dimensional subspace of K^n (where K is a field; and say e_1, …, e_n is the standard basis of K^n) has a canonical basis, where the first e_i to appear in each basis element appears with coefficient 1, and that e_i appears in no other basis element, and that special e_i for that basis element is “to the right” of the e_i of the previous one.  Is there a standard name for this?  (Normal form?  Row-reduced echelon form?)  Is there a good (fairly standard) reference for it?  (Perhaps this gets too far into how linear algebra is taught in different countries, and I should just not give a reference, and instead give it as an exercise.)   (answer here, although I’m also happy to get more references)

8.4.H  Interesting fact:  I almost wanted to say that effective Cartier divisors are the same as codimension 1 regular embeddings.  But I could only show this in the locally Noetherian situation (or more generally, when the structure sheaf is coherent).  The reason for the problem is that the definition of effective Cartier divisor is in terms of open subsets (for good reason), while the definition of regular embedding is in terms of stalks (for good reason), and getting from the latter to the former requires Nakayama.  If you think I’m not giving the right definition of one of these two notions, please let me know!  (see here for an interesting follow-up, thanks to Laurent Moret-Bailly)

9.1.7  Peter Johnson did not like my use of the phrase “open subfunctor” in 9.1.7.  Is  anyone else bothered?  How seriously?  (current plan after discussing with Peter:  leave as is)

9.4.E  Can anyone get this exercise (that, basically, says the product of integral varieties over an algebraically closed field is also an integral variety)?  I think it should be gettable, but no one I know has gotten it (possibly because I haven’t asked it in homework sets).  I want to be sure I have the level correctly gauged.  (Feel free to respond by email if you get stuck.)  (two positive response received so far, including from Gyujin Oh)

10.3.9  Is there an example of a non-smooth group variety over a field k, i.e. a finite type reduced group scheme over k that is not smooth?  Translation:  is there a group variety that is not an algebraic group?  (answer:  yes!  example here)

11.3.13 Over an algebraically closed field, every smooth hypersurface of degree at least n+1 in \mathbb{P}^n is not uniruled.  What is a good reference?  (I know why it is true!  As with many of these questions, I’d like to know where to point people to.) (answer here)

13.8 I mention Tate’s theory of non-archimedean analytic geometry.  Is there a “right” source to point the interested reader (who is just starting out) to?  (possible answer here)

19.9.B  In (, we have j=2^8 (\lambda^2-\lambda+1)^3/(\lambda^2(\lambda-1)^2), and the discussion is away from characteristic 2.  I want to say that the normalization factor 2^8 is because of characteristic 2, but I couldn’t convince myself that this was true.  Presumably it is.  Is there a good reference?  (Remark for comparison:  one can also write j in terms of \tauj = 1728 g_2^3 / \Delta.  Here the prime factors of 1728 are 2 and 3; but the reason for the 3 is not characteristic 3.)  (answer:  yes, see here)

20.2.H  Suppose E is a complex elliptic curve.  Then  \dim_{\mathbb{Q}} N^1_{\mathbb{Q}}(E \times E) is always 3 or 4.  It is 4 if there is a nontrivial endomorphism from E to itself (i.e. not just multiplication by n followed by translation); the additional class comes from the graph of this endomorphism.  Is there a reference for this fact that I can/should direct learners to?   (answer:  yes, see here)

21.5.9   Is there a good reference for the Lefschetz principle?  (Examples currently mentioned:   Kodaira vanishing in characteristic 0;  and non-jumping of hodge numbers in characteristic 0.)  (good answer here)

21.7.8  (not needed)  It is a nontrivial fact that irreducible smooth projective curves of
genus g \geq 2 have finite automorphism groups.  I know three arguments:  using the Neron-Severi theorem (and the Hodge index theorem) (see Hartshorne V.1.9, for example); the fact that the automorphisms are reduced and form a scheme (too hard); and by action on Weierstrass points.  I am surprised that this is so hard.  (Note:  I know that the idea can be quickly outlined to someone learning.  But I want an easy complete rigorous proof.  As long as I am asking, I also want someone to give me a Tesla Roadster.)

21.7.9  Smooth curves in positive characteristic  can have way more than 84(g-1) automorphisms.  Is there a “best” reference?

28.1.L   Is there a canonical reference for Tsen’s theorem, that any proper flat morphism  X \rightarrow Y to a curve, whose geometric fibers are isomorphic to \mathbb{P}^1 is a Zariski \mathbb{P}^1-bundle?  Follow-up question (posted March 5), in response to David Speyer’s comment here:   Does anyone have a (loved) reference for the fact that the universal plane conic (over the space of smooth plane conics) is not a \mathbb{P}^1-bundle?  (See David Speyer’s comment for a little more detail.)  (possible answer to the first question here; answer to the second question here)

29.3.B  I currently define node only in the case of a variety over an algebraically closed field, in which case I say that it is something formally isomorphic to k[[x,y]]/(xy).  I gesture toward the definition in other cases.  For example, if k is not algebraically closed, I define it as k[[x,y]]/q(x,y), where q is a quadratic with no repeated roots.  I want to say that if q is reducible, then this is said to be a split node, and otherwise it is a non-split node.  I’d thought this was standard notation, but google suggests otherwise.  Does anyone have strong feelings about this?

29.5  (This is a follow-up to discussion in the 27th post.)  I am reluctant to introduce new terminology in a well-established field, but there is a notion that I think deserves a name.  Suppose \pi: X \rightarrow Y is a proper morphism.  (For the technically-minded, it is likely that “finitely presented” should also be added, but I will play it safe, and not include this.)  Then I want to say that \pi is [something] if the natural map \mathcal{O}_Y \rightarrow \pi_* \mathcal{O}_X is an isomorphism.   Not EinStein suggested the name \mathcal{O}-connected, and I quite like this — it suggests that this notion is even stronger than connected, and suggests in what way it is stronger.  Another possibility is \mathcal{O}-isomorphic (which  I suggested, but which I currently like less well).  Opinions?  (Are you offended by giving this a new name?  Or do you like one of these suggestions?  Or do you have another idea?)

30.3.4  Is there a canonical (“introductory”)  reference for \pi^! (which will require an introduction to derived categories)?  (Brian Conrad’s book Grothendieck duality and base change perhaps?)  (possible answer here)